Snowflakes

January 18, 2011

Today is another snow day in Northern New Jersey. We've had more snow this season than our usual share, but that's probably just a statistical fluctuation that will be balanced by a milder winter some year in the future. It wouldn't have done us much good to have retreated to the south, since the US southern states have had an unusually severe winter, also. Being stuck in the house, looking out at all that snow, has me thinking about the physical properties of ice, which is what snow really is.

About eight inches of snow fell on Northern New Jersey, January 12, 2011 (Photo by author).

Ice is important to the plot of Kurt Vonnegut's novel, Cat's Cradle. The ice in this case is ice-nine, a fictional phase of ice with a name that's similar to the actual ice phase, ice-IX. Ice-IX is a tetragonal crystalline phase of ice that's stable at very low temperature (< 140 K) and high pressures (200-400 MPa). Ice-IX has a density of 1.16 g/cm3. This is quite unlike the ice outside my window, which has a density of about 0.92, so it floats on water. Fifteen different phases of ice are known, each existing in its own temperature and pressure range. Not surprisingly, the snow outside my window is known as Ice-I. It's even possible to have ice at room temperature and pressure when it's confined to a small dimension.[1-2]

Water and ice have many unusual properties because they're held together by hydrogen bonds. There are so many unusual properties that there are web sites whose sole purpose is to catalog the anomalous properties of water; anomalous, that is, compared to other liquids.[3] One interesting property is the huge change in the high frequency dielectric constant when water freezes. At MHz frequencies and about 0-20oC, water has a dielectric constant of about 80. Ice at its melting point has a dielectric constant of about 3.0 at these frequencies, so there's a huge change in this property in the phase transition from liquid to solid.[4] This fact can lead to useful devices for ice detection.[5]

All this esoteric stuff may be useful, but there's just one thing most people really want to know about snow. Is it really true that no two snowflakes are alike? At one level, typical snowflakes are alike in their hexagonalsymmetry, as first described in 1611 by the astronomer, Johannes Kepler.[6] René Descartes, the Frenchphilosopher and mathematician, made drawings of his naked eye observations of snowflakes, which included some twelve-sided flakes, in 1635.[6] The mathematical Descartes reasoned that snowflakes were hexagonal because they needed to arrange themselves efficiently in clouds. This is likely the first example of tiling by efficient causality.

The greatest advance in snowflake science was the invention of the microscope, which was used by Robert Hooke to examine everything within reach, including snowflakes. His famous 1665 book, Micrographia, included many drawings of snowflakes, and these were the first drawings to indicate the complex shapes within shapes in snowflakes. More than two hundred years later, after the invention of photography, Wilson Alwyn Bentley, a Vermont farmer, took numerous photomicrographs of snowflakes starting in 1885. His photographs documented the fine details inherent in snowflakes and showed how unlikely that two could be the same.

A half century after Bentley started taking his photographs, deuterium was discovered, as were the neutron and isotopes of the elements. At that point, the idea of two snowflakes being exactly alike became absurd, at least if you were an atomist. Natural water, as well as ice formed from that water, has a deuterium atom substituting for one of the hydrogen atoms in every 5000th molecule. The odds of finding a water molecule with 18O substituting for the more common 16O is about one in 500. It doesn't take much mathematics to conclude that various arrangements of these in the 1019 water molecules in a snowflake[7] is a very large number.[6] You would think that 1019 is a large enough number without talking about isotopes, but it's estimated that 1034 snowflakes have fallen in the history of the Earth.[7]

Some countries take their snow very seriously, for the obvious reason that they are covered in it for most of the year. Sweden has issued a set of stamps based on snowflake photographs by Caltechphysicist, Kenneth G. Libbrecht. Libbrecht, who has a snowflake web site,[6] photographed these particular snowflakes in Kiruna, Sweden, in an area known as Lapland (Santareindeer territory).

There's a mathematical object known as the Koch Snowflake. It's a fractal curve invented by the Swedish mathematician, Helge von Koch. What's surprising to people who think that fractals are a modern invention, Koch described this curve in a 1904 paper. It has fractal dimension log 4/log 3, or about 1.26. The snowflake, as shown in the figure, is simply constructed, as follows:[9]

• Draw an equilateral triangle.
• Taking each line segment in turn, divide it into three equal segments.
• Draw an equilateral triangle pointing outwards using the middle segment as the base.
• Remove the base.
• Continue this operation for the next two line segments of the original triangle; and then recursively for all line segments.

There may be applications for such strange objects as the Koch Snowflake. There's a Koch Snowflake Antenna with interesting broadband properties.[10]